Poisson Structure and Moyal Quantisation of the Liouville Theory

TL;DR

This paper explores the Poisson and quantum structures of Liouville theory, deriving its symplectic form, non-equal time brackets, and *-product quantization, linking classical and quantum descriptions.

Contribution

It introduces a gauge-invariant Hamiltonian reduction approach to Liouville theory's symplectic and Poisson structures, and constructs quantum deformations consistent with canonical quantization.

Findings

Derived Poisson brackets for Liouville fields.

Constructed symbols of chiral fields and their *-products.

Presented quantum deformations and non-equal time commutators.

Abstract

The symplectic and Poisson structures of the Liouville theory are derived from the symplectic form of the SL(2,R) WZNW theory by gauge invariant Hamiltonian reduction. Causal non-equal time Poisson brackets for a Liouville field are presented. Using the symmetries of the Liouville theory, symbols of chiral fields are constructed and their *-products calculated. Quantum deformations consistent with the canonical quantisation result, and a non-equal time commutator is given.